I was wondering if it is possible to make any economic implications by regressing the same set of independent variables with different outcome variables.

For instance, regressing [Industry, Years, Production Level] on [Revenue, Profits].

Given the above variables, if both models are significant, what would be the implications? Does it imply that [Industry, Years, Production Level] can be used to explained both the change of [Revenue, Profits].

If it is, what are the appropriate regression models? Are there any concerned issues regarding the results?

$\begingroup$I think both models will be significant because given modest scalability Production Level will correlate with Revenue and given some nearly constant profit margin Profits will correlate Revenue.$\endgroup$
– GiskardJun 3 '15 at 6:03

The interesting realization comes at this point: given the postulated relation $(1)$ it is unlikely that the variables in $\mathbf X$ will be uncorrelated with $\mathbf \varepsilon$, the error term in $(3)$.

This means that even with a large sample, calculating the ratios of the individual coefficients in the vectors $\mathbf {\hat a},\;\mathbf {\hat b}$, we will not recover $\gamma$, since $\left(\mathbf X'\mathbf X\right)^{-1}\mathbf X'\mathbf \varepsilon$ won't converge to zero.